In many countries they don't accept kids to nursery without up to date vacinnation card.

This is true in the U.S. as well [0]. Some states allow for religious, philosophical, or medical exemptions; these exceptions are similar to the systems implemented in the Netherlands (religious [1]), Australia (philosophical [2]) or Germany (medical [1]).

[0]: https://en.wikipedia.org/wiki/Vaccination_policy_in_the_Unit...

[1]: https://link.springer.com/content/pdf/10.1057/s41271-019-001...

[2]: https://www.sabin.org/sites/sabin.org/files/restricted/Salmo...

They also don’t allow you to collect welfare unless you and your kids have received your vaccinations (Australia).

I have travelled South East Asia for several years and never been asked to show any proof.